Paul's "T Parity Notes"
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Paul's "T Parity Notes"
es2mac Wed Apr 04, 2012 10:39 pm
Some time ago I summarized what I knew about the "parity" properties of the T tetromino, into a set of fumens. Later I added some explanations so that others can follow, and here it is. It's sort of long, but the message is simple: T is special, handle with care! Questions and comments welcome.
This note was originally just the 11 fumens I drew up as personal notes on the topic, expanded into current form on a whim.
Among the 7 tetrominos, the T piece has a unique property in terms of polyominoes tiling. Other than theoretical interests, any good player would already have an intuitive understanding of its implications on tetris stacking. Here things tend to be related to certain numbers being even or odd, which is referred to as "parity." For convenience (laziness) sake I also casually use the word parity to refer to the property of T, or a certain state of a tetris stack. Which gets pretty bad, but before a better term is coined anyways.
Say we color our matrix with alternating black and white cells (minos) like a chess board, and place a tetromino on the board. Any tetromino other than T would cover two blacks and two whites (even, and same, numbers), but T would cover either one black three whites, or one white three blacks (odd, and different, numbers). Rotating the T doesn't change how many blacks and whites it cover, so we can say that the "state" of cell covering of the T piece is determined by the location of its center, invariant of rotation.
1. 7 minos cover diagram.
http://tinyurl.com/cgodfuq
Say we have two T's and we want to know if they are in complementary positions, that is, one covers more blacks while the other covers more whites, so together they cover the same number of blacks and whites. This is easily done by counting the distance between their centers. They are complements if and only if the distance between centers is odd.
2. T center distance counting. When two Ts have centers that are odd cells distance apart, their centers will be of opposite colors, so they are complements.
http://tinyurl.com/coan4ql
Suppose we want to assemble tetrominos into a rectangle shape. First thing we know is that the number of cells in it is a multiple of 4 (because every tetromino takes 4 cells). We also know that when coloring this rectangle with black and white cells, it would have as many blacks as whites. Therefore! For every T used in making this rectangle, there must be another complementing T in there somewhere. Which is to say, we need an even number of T pieces.
3. Rectangle filling. The two Ts used in making up the rectangle must be complements.
http://tinyurl.com/cdmycyb
When we say making a rectangle with tetrominos, one of the first things we'd think of would be the perfect clear opening. The funny thing is that after setting up the first bag, we don't always need a T to complete the perfect clear. The idea is that skimming could change (fix) the parity in a stack.
4. Perfect clear opening, one bag + 3 pieces. If T is not present in the 3 pieces, then we need to skim in some way. So that after skimming, the number of blacks/whites to fill are the same.
http://tinyurl.com/cbe8qp8
Now let's consider parity in terms of normal stacking. Experience tells us that T makes flat stacks jagged, and makes jagged stack flat. If we narrow down to say stacking 4 columns, this is very obvious. But now we know that this is because of the parity property of T, we can safely say that the only fix to a field with parity issue (T-ified field?) without skimming is with another T. A T-ified field is jagged looking, and is only good for placing S, Z or another T.
5. 4-column stacking example. Jaggedness and uneven parity (by which I mean the unbalance between white and black cell counts) go hand in hand.
http://tinyurl.com/cwekofj
The above example is particularly relevant in ST stacking, where on the left side (left of ZT... right of ST) 4 columns need to take T's in pairs. Do the same thing for 3 columns and we have a similar situation, and would be relevant in center gap 4-wide.
If we're stacking wider columns, say tetrising (9 columns), then we'd want to put T's in complementing positions, so we get flat areas for our pieces. Otherwise we'd have a hard time placing O, L or J, and have fewer/worse options for S and Z.
6. Complementing or otherwise, stacking options compared.
http://tinyurl.com/7rxwupv
Luckily we don't have to think much about parity issues when we play, because even though (I would say) parity is a concern, we intuitively recognize fields with even or uneven parity. A field with even parity looks flat and easy to stack on, a field with uneven parity is jagged and... unappetizing.
7. Recognizing the shape of uneven parity. Sawtooth shapes and other unfavorable stacking surfaces tend to have uneven parity.
http://tinyurl.com/bws2ag5
Here's a random run of tetris stacking that has a purposely badly placed T. By counting centers, we'd see that the second T is in non-complementing position with the first, and after that stacking gets pretty awkward. Though for the third T, there is really no way to place it badly again (there's no non-gap-making place to put it that is not in complement with the first two) though it wasn't put where it is needed most anyways.
8. Example of bad T placement (tetris stacking)
http://tinyurl.com/cvycbra
How about the first T? Say we're stacking for tetris with gap on the far right, and T comes early so we're placing it down first. Then perhaps some T placements are better than others... in theory. Personally I like the two positions close to the center in the fumen below.
9. Tetris stacking first T placement. The 9-wide column rectangles tend to have one more white cell than black, so if the first T covers more whites, then perhaps we get a smoother tetris opening. In the last few pages of the fumen, purple cells enclose the T positions that cover more whites.
http://tinyurl.com/cjvu6zk
Of course, since a field with parity issue is begging for a T, it tends to be ideal for T-spins.
10. T-spins. Our field has uneven parity, is hard to stack on, and is begging for Ts to fix things up. Might as well make a T-spin. Even before our T comes, we get flat tops to work with, lovely.
http://tinyurl.com/d2u9atb
Finally, just like in the perfect clear case, skimming (esp. with L/J) often fixes parity issue in situations where no T is available. This is useful when downstacking, dig challenge etc.
11. Two skimming fix examples. Skimming could reduce T dependence. Also imagine in these two cases, how jagged the field would look if we don't skim or use a T.
http://tinyurl.com/c9mwydf
Reference
http://waka.nu/tetris/parity/[list=1][*]
This note was originally just the 11 fumens I drew up as personal notes on the topic, expanded into current form on a whim.
Among the 7 tetrominos, the T piece has a unique property in terms of polyominoes tiling. Other than theoretical interests, any good player would already have an intuitive understanding of its implications on tetris stacking. Here things tend to be related to certain numbers being even or odd, which is referred to as "parity." For convenience (laziness) sake I also casually use the word parity to refer to the property of T, or a certain state of a tetris stack. Which gets pretty bad, but before a better term is coined anyways.
Say we color our matrix with alternating black and white cells (minos) like a chess board, and place a tetromino on the board. Any tetromino other than T would cover two blacks and two whites (even, and same, numbers), but T would cover either one black three whites, or one white three blacks (odd, and different, numbers). Rotating the T doesn't change how many blacks and whites it cover, so we can say that the "state" of cell covering of the T piece is determined by the location of its center, invariant of rotation.
1. 7 minos cover diagram.
http://tinyurl.com/cgodfuq
Say we have two T's and we want to know if they are in complementary positions, that is, one covers more blacks while the other covers more whites, so together they cover the same number of blacks and whites. This is easily done by counting the distance between their centers. They are complements if and only if the distance between centers is odd.
2. T center distance counting. When two Ts have centers that are odd cells distance apart, their centers will be of opposite colors, so they are complements.
http://tinyurl.com/coan4ql
Suppose we want to assemble tetrominos into a rectangle shape. First thing we know is that the number of cells in it is a multiple of 4 (because every tetromino takes 4 cells). We also know that when coloring this rectangle with black and white cells, it would have as many blacks as whites. Therefore! For every T used in making this rectangle, there must be another complementing T in there somewhere. Which is to say, we need an even number of T pieces.
3. Rectangle filling. The two Ts used in making up the rectangle must be complements.
http://tinyurl.com/cdmycyb
When we say making a rectangle with tetrominos, one of the first things we'd think of would be the perfect clear opening. The funny thing is that after setting up the first bag, we don't always need a T to complete the perfect clear. The idea is that skimming could change (fix) the parity in a stack.
4. Perfect clear opening, one bag + 3 pieces. If T is not present in the 3 pieces, then we need to skim in some way. So that after skimming, the number of blacks/whites to fill are the same.
http://tinyurl.com/cbe8qp8
Now let's consider parity in terms of normal stacking. Experience tells us that T makes flat stacks jagged, and makes jagged stack flat. If we narrow down to say stacking 4 columns, this is very obvious. But now we know that this is because of the parity property of T, we can safely say that the only fix to a field with parity issue (T-ified field?) without skimming is with another T. A T-ified field is jagged looking, and is only good for placing S, Z or another T.
5. 4-column stacking example. Jaggedness and uneven parity (by which I mean the unbalance between white and black cell counts) go hand in hand.
http://tinyurl.com/cwekofj
The above example is particularly relevant in ST stacking, where on the left side (left of ZT... right of ST) 4 columns need to take T's in pairs. Do the same thing for 3 columns and we have a similar situation, and would be relevant in center gap 4-wide.
If we're stacking wider columns, say tetrising (9 columns), then we'd want to put T's in complementing positions, so we get flat areas for our pieces. Otherwise we'd have a hard time placing O, L or J, and have fewer/worse options for S and Z.
6. Complementing or otherwise, stacking options compared.
http://tinyurl.com/7rxwupv
Luckily we don't have to think much about parity issues when we play, because even though (I would say) parity is a concern, we intuitively recognize fields with even or uneven parity. A field with even parity looks flat and easy to stack on, a field with uneven parity is jagged and... unappetizing.
7. Recognizing the shape of uneven parity. Sawtooth shapes and other unfavorable stacking surfaces tend to have uneven parity.
http://tinyurl.com/bws2ag5
Here's a random run of tetris stacking that has a purposely badly placed T. By counting centers, we'd see that the second T is in non-complementing position with the first, and after that stacking gets pretty awkward. Though for the third T, there is really no way to place it badly again (there's no non-gap-making place to put it that is not in complement with the first two) though it wasn't put where it is needed most anyways.
8. Example of bad T placement (tetris stacking)
http://tinyurl.com/cvycbra
How about the first T? Say we're stacking for tetris with gap on the far right, and T comes early so we're placing it down first. Then perhaps some T placements are better than others... in theory. Personally I like the two positions close to the center in the fumen below.
9. Tetris stacking first T placement. The 9-wide column rectangles tend to have one more white cell than black, so if the first T covers more whites, then perhaps we get a smoother tetris opening. In the last few pages of the fumen, purple cells enclose the T positions that cover more whites.
http://tinyurl.com/cjvu6zk
Of course, since a field with parity issue is begging for a T, it tends to be ideal for T-spins.
10. T-spins. Our field has uneven parity, is hard to stack on, and is begging for Ts to fix things up. Might as well make a T-spin. Even before our T comes, we get flat tops to work with, lovely.
http://tinyurl.com/d2u9atb
Finally, just like in the perfect clear case, skimming (esp. with L/J) often fixes parity issue in situations where no T is available. This is useful when downstacking, dig challenge etc.
11. Two skimming fix examples. Skimming could reduce T dependence. Also imagine in these two cases, how jagged the field would look if we don't skim or use a T.
http://tinyurl.com/c9mwydf
Reference
http://waka.nu/tetris/parity/[list=1][*]
es2mac- Posts : 42
Join date : 2012-03-23
Location : New York -
Re: Paul's "T Parity Notes"
iHotShot Wed Apr 04, 2012 10:45 pm
I was reading the third paragraph and I thought, oh yeah..
Good stuff man.
Good stuff man.
iHotShot- Posts : 24
Join date : 2012-03-25
Age : 33
Location : Philippines
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